Degree of recurrence of generic diffeomorphisms

TL;DR

This paper investigates how well numerical discretizations of generic conservative diffeomorphisms on the torus preserve dynamical features, revealing significant differences between smooth and continuous regularity cases.

Contribution

It demonstrates that discretizations of $C^r$ generic conservative diffeomorphisms exhibit fundamentally different dynamics from the $C^0$ case, using new local-global formulas and quasicrystal theory.

Findings

Discretizations of $C^r$ diffeomorphisms differ from $C^0$ dynamics.

A local-global formula for discretizations is established.

Linear case analysis employs quasicrystal concepts.

Abstract

We study spatial discretizations of dynamical systems: is it possible to recover some dynamical features of a system from numerical simulations? Here, we tackle this issue for the simplest algorithm possible: we compute long segments of orbits with a fixed number of digits. We show that for every $r\ge 1$, the dynamics of the discretizations of a $C^r$ generic conservative diffeomorphism of the torus is very different from that observed in the $C^0$ regularity. The proof of our results involves in particular a local-global formula for discretizations, as well as a study of the corresponding linear case, which uses ideas from the theory of quasicrystals.